1. Static Fluids Fluids are substances, such as liquids and gases, that have no rigidity. A fluid lacks a fixed shape and assumes the shape of its container. Fluids are substances, such as liquids and gases, that have no rigidity. A fluid lacks a fixed shape and assumes the shape of its container. In the liquid state, molecules can flow; they freely move from position to position by sliding over one another. In the liquid state, molecules can flow; they freely move from position to position by sliding over one another.

2. Pressure The pressure P acting on a fluid is the force exerted perpendicularly per unit of the fluid’s surface area The pressure P acting on a fluid is the force exerted perpendicularly per unit of the fluid’s surface area Unit of pressure is the N/m 2 or Pascal; 1 N/m 2 = 1 Pa (Pascal). Unit of pressure is the N/m 2 or Pascal; 1 N/m 2 = 1 Pa (Pascal). Atmospheric pressure at sea level is 1 atmosphere (atm) = 1.013 x 10 5 Pa. Atmospheric pressure at sea level is 1 atmosphere (atm) = 1.013 x 10 5 Pa. 1 atm = 14.7 lb/in 2. 1 atm = 14.7 lb/in 2.

3. Pressure in a Liquid. A liquid in a container exerts forces against the walls and bottom of the container. A liquid in a container exerts forces against the walls and bottom of the container. For a liquid in a container, the pressure the liquid exerts against the bottom of the container is the weight of the liquid divided by the area of the container bottom. For a liquid in a container, the pressure the liquid exerts against the bottom of the container is the weight of the liquid divided by the area of the container bottom.

4. Density Density D of a substance is its mass per unit volume ; Density D of a substance is its mass per unit volume ; Objects composed of the same substance, whatever the size or mass, have the same density under the same conditions of temperature and pressure. Objects composed of the same substance, whatever the size or mass, have the same density under the same conditions of temperature and pressure. Temperature and pressure affect the density of substances, appreciably for gases, but only slightly for liquids and solids. Temperature and pressure affect the density of substances, appreciably for gases, but only slightly for liquids and solids.

5. Density How much a liquid weighs and how much pressure it exerts depends on its density. How much a liquid weighs and how much pressure it exerts depends on its density. – For the same depth, a denser liquid exerts a greater pressure than a less dense liquid. – For liquids of the same density, the pressure will be greater at the bottom of the deeper liquid.

6. Columnar Fluid Pressure (sometimes called gauge pressure) pressure due to a column of fluid of height h and mass density D; pressure due to a column of fluid of height h and mass density D; The pressure of a liquid at rest depends on the density and depth of the liquid. The pressure of a liquid at rest depends on the density and depth of the liquid. Liquids are practically incompressible, so except for changes in the temperature, the density of a liquid is normally the same at all depths. Liquids are practically incompressible, so except for changes in the temperature, the density of a liquid is normally the same at all depths.

7. Columnar Fluid Pressure At a given depth, a given liquid exerts the same pressure against any surface - the bottom or sides of its container, or even the surface of an object submerged in the liquid to that depth. At a given depth, a given liquid exerts the same pressure against any surface - the bottom or sides of its container, or even the surface of an object submerged in the liquid to that depth. Pressure a liquid exerts depends only on its density and depth. Pressure a liquid exerts depends only on its density and depth. Total pressure (or absolute pressure) P on a submerged surface equals the pressure the liquid exerts plus the atmospheric pressure P o ( 1 atm = 1.013 x 10 5 Pa). Total pressure (or absolute pressure) P on a submerged surface equals the pressure the liquid exerts plus the atmospheric pressure P o ( 1 atm = 1.013 x 10 5 Pa).

8. Fluid Pressure Pressure of a liquid does not depend on the amount of liquid. Pressure of a liquid does not depend on the amount of liquid. Neither the volume or total weight of the liquid matters. Neither the volume or total weight of the liquid matters. If you sampled water pressure at 1 m beneath a large lake surface and 1 m beneath a small pool surface, the pressure would be the same. If you sampled water pressure at 1 m beneath a large lake surface and 1 m beneath a small pool surface, the pressure would be the same. The fact that water pressure depends on depth and not on volume is illustrated by Pascal vases. The fact that water pressure depends on depth and not on volume is illustrated by Pascal vases. Water surface in each of the connected vases is at the same level. Water surface in each of the connected vases is at the same level. Occurs because the pressures at equal depths beneath the surface are the same. Occurs because the pressures at equal depths beneath the surface are the same.

9. Fluid Pressure At any point within a liquid, the forces that produce pressure are exerted equally in all directions. At any point within a liquid, the forces that produce pressure are exerted equally in all directions.

10. Forces Exerted By a Fluid When the liquid is pressing against a surface there is a net force directed perpendicular to the surface. When the liquid is pressing against a surface there is a net force directed perpendicular to the surface. If there is a hole in the surface, the liquid initially moves perpendicular to the surface. If there is a hole in the surface, the liquid initially moves perpendicular to the surface. At greater depths, the net force is greater and the horizontal velocity of the escaping liquid is greater. At greater depths, the net force is greater and the horizontal velocity of the escaping liquid is greater.

11. Transmission of Pressure: Pascal’s Principle. Pascal’s Principle: A CHANGE IN PRESSURE IN A CONFINED FLUID IS TRANSMITTED WITHOUT CHANGE TO ALL POINTS IN THE FLUID. Pascal’s Principle: A CHANGE IN PRESSURE IN A CONFINED FLUID IS TRANSMITTED WITHOUT CHANGE TO ALL POINTS IN THE FLUID. Ex. Hydraulic lift. Ex. Hydraulic lift. Hydraulic piston apparatus uses an incompressible fluid to transmit pressure from a small cylinder to a large cylinder. Hydraulic piston apparatus uses an incompressible fluid to transmit pressure from a small cylinder to a large cylinder. According to Pascal’s Principle, the pressure in the small cylinder resulting from the application of F 1 to a frictionless piston is transmitted undiminished to the larger piston. According to Pascal’s Principle, the pressure in the small cylinder resulting from the application of F 1 to a frictionless piston is transmitted undiminished to the larger piston.

12. Transmission of pressure: Pascal’s Principle. P 1 = P 2 A 2 is larger than A 1, so the force exerted by the large piston is greater than the force exerted on the small piston. A 2 is larger than A 1, so the force exerted by the large piston is greater than the force exerted on the small piston. AMA (actual mechanical advantage) for hydraulic lift: AMA (actual mechanical advantage) for hydraulic lift:

13. Transmission of pressure: Pascal’s Principle. The figure shows a hydraulic system used with brakes. The force F is applied perpendicularly to the brake pedal. The brake pedal rotates about the axis shown in the drawing and causes a force to be applied perpendicularly to the input piston in the master cylinder. The resulting pressure is transmitted by the brake fluid to the output plungers which are covered with the brake linings. The linings are pressed against both sides of a disc attached to the rotating wheel. The figure shows a hydraulic system used with brakes. The force F is applied perpendicularly to the brake pedal. The brake pedal rotates about the axis shown in the drawing and causes a force to be applied perpendicularly to the input piston in the master cylinder. The resulting pressure is transmitted by the brake fluid to the output plungers which are covered with the brake linings. The linings are pressed against both sides of a disc attached to the rotating wheel.

14. Buoyancy Buoyancy: the apparent loss of weight of objects when submerged in a liquid. Buoyancy: the apparent loss of weight of objects when submerged in a liquid. Easier to lift objects under water surface than to lift it above the water surface. Easier to lift objects under water surface than to lift it above the water surface. When submerged, water exerts an upward force that is opposite in direction to gravity. Upward force called the buoyant force. When submerged, water exerts an upward force that is opposite in direction to gravity. Upward force called the buoyant force.

15. Buoyancy Forces exerted by liquid produce pressure against the submerged object. Forces exerted by liquid produce pressure against the submerged object. Forces are greater at greater depths; forces are equal at the same depth on opposite sides of the object. Forces are greater at greater depths; forces are equal at the same depth on opposite sides of the object. Forces acting upward on the bottom of the object greater than those acting downward on top of the object, simply because the bottom of the object is deeper. Forces acting upward on the bottom of the object greater than those acting downward on top of the object, simply because the bottom of the object is deeper. Difference in upward and downward forces is the buoyant force, B. Difference in upward and downward forces is the buoyant force, B. Fs refers to the force the scale exerts on the mass m. You may also refer to Fs as the tension in a supporting string. Fs refers to the force the scale exerts on the mass m. You may also refer to Fs as the tension in a supporting string.

16. Buoyancy If the weight of the object is greater than the buoyant force, the object will sink (as in figure a). If the weight of the object is greater than the buoyant force, the object will sink (as in figure a). If the weight of the object is equal to the buoyant force, the net force on the object is zero and the submerged object will remain at any level (as in figure b). If the weight of the object is equal to the buoyant force, the net force on the object is zero and the submerged object will remain at any level (as in figure b). If the weight of the object is less than the buoyant force, the object will rise to the surface and float. If the weight of the object is less than the buoyant force, the object will rise to the surface and float.

17. Buoyancy When an object is submerged in a liquid, the liquid level will rise. When an object is submerged in a liquid, the liquid level will rise. Liquid is displaced or moved elsewhere. Liquid is displaced or moved elsewhere. The volume of the liquid displaced is equal to the volume of the submerged object. The volume of the liquid displaced is equal to the volume of the submerged object. A completely submerged object always displaces a volume of liquid equal to its own volume. A completely submerged object always displaces a volume of liquid equal to its own volume.

18. Archimede’s Principle When an object is immersed in a fluid, it appears to weigh less. When an object is immersed in a fluid, it appears to weigh less. Archimede’s Principle: THE BUOYANT FORCE EXERTED ON A BODY WHOLLY OR PARTLY IMMERSED IN A FLUID IS EQUAL TO THE WEIGHT OF THE FLUID DISPLACED BY THE BODY. Archimede’s Principle: THE BUOYANT FORCE EXERTED ON A BODY WHOLLY OR PARTLY IMMERSED IN A FLUID IS EQUAL TO THE WEIGHT OF THE FLUID DISPLACED BY THE BODY. Archimede’s Principle applies to both liquids and gases, which are fluids. Archimede’s Principle applies to both liquids and gases, which are fluids. Immersed refers to either completely or partially submerged. Immersed refers to either completely or partially submerged.

19. Archimede’s Principle Buoyant force: upward force on the object when it is immersed in water. Buoyant force: upward force on the object when it is immersed in water. The buoyant force (BF) is the weight of the displaced fluid - not the weight of the submerged object. The buoyant force (BF) is the weight of the displaced fluid - not the weight of the submerged object. BF = (mass in air - mass in fluid)·gravity BF = (mass in air - mass in fluid)·gravity BF = weight of displaced fluid = m·g = D·V·g BF = weight of displaced fluid = m·g = D·V·g Apparent weight of a submerged object is its weight in air minus the buoyant force. Apparent weight of a submerged object is its weight in air minus the buoyant force. Apparent weight = m·g – BF = m·g - D·V·g

20. Flotation Principle of Flotation: A FLOATING OBJECT DISPLACES A WEIGHT OF FLUID EQUAL TO ITS OWN WEIGHT. Principle of Flotation: A FLOATING OBJECT DISPLACES A WEIGHT OF FLUID EQUAL TO ITS OWN WEIGHT. A simple relationship between the weight of a submerged object and the buoyant force can be found by considering their ratio: A simple relationship between the weight of a submerged object and the buoyant force can be found by considering their ratio:

21. Flotation Shipbuilding and the principle of flotation: Shipbuilding and the principle of flotation: – A solid 1-ton block of iron is nearly 8 times as dense as water, so when it is submerged, it will displace 1/8 ton of water (not an amount equal to 8 tons of water). – Reshape the iron block into a bowl and submerge, displaces a greater volume of water. The deeper the bowl is immersed at the surface, the more water is displaced and the greater is the buoyant force exerted on the bowl.

22. Flotation When the weight of the displaced water equals the weight of the bowl - flotation. Flotation occurs when the weight of the bowl equals the buoyant force. When the weight of the displaced water equals the weight of the bowl - flotation. Flotation occurs when the weight of the bowl equals the buoyant force. Every ship must be designed to displace a weight of water equal to its own weight. Every ship must be designed to displace a weight of water equal to its own weight. Submarines: Submarines: – Displace a weight of water equal to its own weight, it remains at a constant depth. – Displaces a weight of water greater than its own weight, rises. – Displaces a weight of water less than its own weight, sinks.

23. Buoyancy in Two Liquids of Differing Density If you have an object submerged in two liquids of different density, such that the upper portion of the object is located in the upper liquid and the lower portion of the object is located in the lower liquid, the total buoyant force on the object is equal to the weight of the object (BF = D·V·g). If you have an object submerged in two liquids of different density, such that the upper portion of the object is located in the upper liquid and the lower portion of the object is located in the lower liquid, the total buoyant force on the object is equal to the weight of the object (BF = D·V·g). Ex. A piece of wood floating partially in water and partially in oil. The density of the oil is less than the density of the water. Ex. A piece of wood floating partially in water and partially in oil. The density of the oil is less than the density of the water. If d is the height of the object, let y = the portion of the object in the more dense liquid and d-y = the portion of the object in the less dense liquid. If d is the height of the object, let y = the portion of the object in the more dense liquid and d-y = the portion of the object in the less dense liquid.

24. Buoyancy in Two Liquids of Differing Density Gravity cancels out. Gravity cancels out. This equation can then be solved for the unknown variable. This equation can then be solved for the unknown variable.

25. Pressure Example Water is to be pumped to the top of the Empire State Building, which is 366 m high. What gauge pressure is needed to raise the water to a height of 366 m? The density of water is 1000 kg/m 3. Water is to be pumped to the top of the Empire State Building, which is 366 m high. What gauge pressure is needed to raise the water to a height of 366 m? The density of water is 1000 kg/m 3.

26. Buoyant Force An object weighing 300 N in air is immersed in water after being tied to a string connected to a balance. The scale now reads 265 N. Immersed in oil, the object appears to weigh 275 N. An object weighing 300 N in air is immersed in water after being tied to a string connected to a balance. The scale now reads 265 N. Immersed in oil, the object appears to weigh 275 N. A. Find the density of the object. A. Find the density of the object.

27. Buoyant Force B. Determine the density of the oil. B. Determine the density of the oil.